Proportional Editing Falloff
Warp Modifier with Texture of your choice including sinusoidal.

There exists three or more modifiers that are relevant.


the wave modifier 
the warp modifier which can take a texture such as a sine texture.
the displace modifier


Other techniques


weight painting and weight modifiers
texture to affect modifiers




Above is the warp modifier in action.  Rather plain ... no texture.



Closeup Fall Off with texture. Image Above.



Warp Modifier with Wood Sine Texture. Image above.

The warp modifier produces results similar to user gestures of 

proportional editing.

The blue cube has a high vertex density.  The from and to are animated in rotation.




Results of the wave modifier. Sinusoidal influence. Image Above.